New Maximal Gaps - Page 8

rudy235 · 2023-02-23, 23:27

Quote:

Originally Posted by Bobby Jacobs

1370 is the 73rd maximal prime gap. We really want to know if 1552 is the 81st maximal prime gap.

Yes

mart_r · 2023-02-24, 19:41

Perhaps it would be expedient to consider a complete trial division sieve on large intervals in order for the search to remain exhaustive.

Not sure what's possible in terms of efficient computation, but I'm envisioning a large bitmap, for example 32 GB in size, which could contain almost 3e12 numbers if a wheel size of 30030 is used and only numbers of the form 6k+1 are being looked at (a gap of >= 1432 should appear every about 5e8 integers, only those would have to be checked again for 6k-1 in-between).

To compete, we expect a throughput of at least, say, 3e10 per second. Trial division needs about 210 million primes (i.e. on top about 200 MB to store the halved differences between primes, one per byte), where I guess memory writing is the bottleneck for small primes, "mod"-ing for large primes may be similarly slow (a simplification, I know, there's certainly more to it). Searching for gaps in the sieved interval is the next possible bottleneck, I don't know how much can be done in parallel.
If one such interval could be dealt with in less than two minutes, we'd be happy.

Is this worth any thought? What are the most critical hurdles we are facing?

Bobby Jacobs · 2023-03-05, 17:14

Yes. This is a good idea. It does not seem to have any hurdles. You did primes up to 2⁶⁴ quickly, so this should be quick, too.

mart_r · 2023-03-06, 19:01

Quote:

Originally Posted by Bobby Jacobs

You did primes up to 2⁶⁴ quickly,

Me? Not so much, my performance as a participant back in 2018 was only mediocre, remember that I didn't find any first occurrences. Unlucky me

Also I don't know how to write code comparable to that of Robert Gerbicz's. It takes way more skill and dexterity than a few lines of Pari or Basic...

Andrew Usher · 2023-03-20, 12:30

The obvious question is: to what extent can that code be re-used in a search beyond 2^64? Also, because of the smaller interval required (less than 2^55), it doesn't need to be as efficient.

2023-02-24, 19:41	#79
mart_r Dec 2008 you know...around... 2³×109 Posts	SOE all over again? Perhaps it would be expedient to consider a complete trial division sieve on large intervals in order for the search to remain exhaustive. Not sure what's possible in terms of efficient computation, but I'm envisioning a large bitmap, for example 32 GB in size, which could contain almost 3e12 numbers if a wheel size of 30030 is used and only numbers of the form 6k+1 are being looked at (a gap of >= 1432 should appear every about 5e8 integers, only those would have to be checked again for 6k-1 in-between). To compete, we expect a throughput of at least, say, 3e10 per second. Trial division needs about 210 million primes (i.e. on top about 200 MB to store the halved differences between primes, one per byte), where I guess memory writing is the bottleneck for small primes, "mod"-ing for large primes may be similarly slow (a simplification, I know, there's certainly more to it). Searching for gaps in the sieved interval is the next possible bottleneck, I don't know how much can be done in parallel. If one such interval could be dealt with in less than two minutes, we'd be happy. Is this worth any thought? What are the most critical hurdles we are facing?

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2023-03-05, 17:14	#80
Bobby Jacobs May 2018 3·5·19 Posts	Yes. This is a good idea. It does not seem to have any hurdles. You did primes up to 2⁶⁴ quickly, so this should be quick, too.

2023-03-20, 12:30	#82
Andrew Usher Dec 2022 2·3²·17 Posts	The obvious question is: to what extent can that code be re-used in a search beyond 2^64? Also, because of the smaller interval required (less than 2^55), it doesn't need to be as efficient.